Improvement to an existing multi-level capacitated lot sizing problem considering setup carryover, backlogging, and emission control

This paper presents a multi-level, multi-item, multi-period capacitated lot-sizing problem. The lot-sizing problem studies can obtain production quantities, setup decisions and inventory levels in each period fulfilling the demand requirements with limited capacity resources, considering the Bill of Material (BOM) structure while simultaneously minimizing the production, inventory, and machine setup costs. The paper proposes an exact solution to Chowdhury et al. (2018)\u27s[1] developed model, which considers the backlogging cost, setup carryover & greenhouse gas emission control to its model complexity. The problem contemplates the Dantzig-Wolfe (D.W.) decomposition to decompose the multi-level capacitated problem into a single-item uncapacitated lot-sizing sub-problem. To avoid the infeasibilities of the weighted problem (WP), an artificial variable is introduced, and the Big-M method is employed in the D.W. decomposition to produce an always feasible master problem. In addition, Wagner & Whitin\u27s[2] forward recursion algorithm is also incorporated in the solution approach for both end and component items to provide the minimum cost production plan. Introducing artificial variables in the D.W. decomposition method is a novel approach to solving the MLCLSP model. A better performance was achieved regarding reduced computational time (reduced by 50%) and optimality gap (reduced by 97.3%) in comparison to Chowdhury et al. (2018)\u27s[1] developed model

Lot-Sizing Problem for a Multi-Item Multi-level Capacitated Batch Production System with Setup Carryover, Emission Control and Backlogging using a Dynamic Program and Decomposition Heuristic

Wagner and Whitin (1958) develop an algorithm to solve the dynamic Economic Lot-Sizing Problem (ELSP), which is widely applied in inventory control, production planning, and capacity planning. The original algorithm runs in O(T^2) time, where T is the number of periods of the problem instance. Afterward few linear-time algorithms have been developed to solve the Wagner-Whitin (WW) lot-sizing problem; examples include the ELSP and equivalent Single Machine Batch-Sizing Problem (SMBSP). This dissertation revisits the algorithms for ELSPs and SMBSPs under WW cost structure, presents a new efficient linear-time algorithm, and compares the developed algorithm against comparable ones in the literature. The developed algorithm employs both lists and stacks data structure, which is completely a different approach than the rest of the algorithms for ELSPs and SMBSPs. Analysis of the developed algorithm shows that it executes fewer number of basic actions throughout the algorithm and hence it improves the CPU time by a maximum of 51.40% for ELSPs and 29.03% for SMBSPs. It can be concluded that the new algorithm is faster than existing algorithms for both ELSPs and SMBSPs. Lot-sizing decisions are crucial because these decisions help the manufacturer determine the quantity and time to produce an item with a minimum cost. The efficiency and productivity of a system is completely dependent upon the right choice of lot-sizes. Therefore, developing and improving solution procedures for lot-sizing problems is key. This dissertation addresses the classical Multi-Level Capacitated Lot-Sizing Problem (MLCLSP) and an extension of the MLCLSP with a Setup Carryover, Backlogging and Emission control. An item Dantzig Wolfe (DW) decomposition technique with an embedded Column Generation (CG) procedure is used to solve the problem. The original problem is decomposed into a master problem and a number of subproblems, which are solved using dynamic programming approach. Since the subproblems are solved independently, the solution of the subproblems often becomes infeasible for the master problem. A multi-step iterative Capacity Allocation (CA) heuristic is used to tackle this infeasibility. A Linear Programming (LP) based improvement procedure is used to refine the solutions obtained from the heuristic method. A comparative study of the proposed heuristic for the first problem (MLCLSP) is conducted and the results demonstrate that the proposed heuristic provide less optimality gap in comparison with that obtained in the literature. The Setup Carryover Assignment Problem (SCAP), which consists of determining the setup carryover plan of multiple items for a given lot-size over a finite planning horizon is modelled as a problem of finding Maximum Weighted Independent Set (MWIS) in a chain of cliques. The SCAP is formulated using a clique constraint and it is proved that the incidence matrix of the SCAP has totally unimodular structure and the LP relaxation of the proposed SCAP formulation always provides integer optimum solution. Moreover, an alternative proof that the relaxed ILP guarantees integer solution is presented in this dissertation. Thus, the SCAP and the special case of the MWIS in a chain of cliques are solvable in polynomial time

An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging

This paper proposes two new mixed integer programming models for capacitated multi-level lot-sizing problems with backlogging, whose linear programming relaxations provide good lower bounds on the optimal solution value. We show that both of these strong formulations yield the same lower bounds. In addition to these theoretical results, we propose a new, effective optimization framework that achieves high quality solutions in reasonable computational time. Computational results show that the proposed optimization framework is superior to other well-known approaches on several important performance dimensions

Meta-Heuristics for Dynamic Lot Sizing: a review and comparison of solution approaches

Proofs from complexity theory as well as computational experiments indicate that most lot sizing problems are hard to solve. Because these problems are so difficult, various solution techniques have been proposed to solve them. In the past decade, meta-heuristics such as tabu search, genetic algorithms and simulated annealing, have become popular and efficient tools for solving hard combinational optimization problems. We review the various meta-heuristics that have been specifically developed to solve lot sizing problems, discussing their main components such as representation, evaluation neighborhood definition and genetic operators. Further, we briefly review other solution approaches, such as dynamic programming, cutting planes, Dantzig-Wolfe decomposition, Lagrange relaxation and dedicated heuristics. This allows us to compare these techniques. Understanding their respective advantages and disadvantages gives insight into how we can integrate elements from several solution approaches into more powerful hybrid algorithms. Finally, we discuss general guidelines for computational experiments and illustrate these with several examples

DYNAMIC LOT-SIZING PROBLEMS: A Review on Model and Efficient Algorithm

Due to their importance in industry, dynamic demand lot-sizing problems are frequently studied.This study consider dynamic lot-sizing problems with recent advances in problem and modelformulation, and algorithms that enable large-scale problems to be effectively solved.Comprehensive review is given on model formulation of dynamic lot-sizing problems, especiallyon capacitated lot-sizing (CLS) problem and the coordinated lot-sizing problem. Bothapproaches have their intercorrelated, where CLS can be employed for single or multilevel/stage, item, and some restrictions. When a need for joint setup replenishment exists, thenthe coordinated lot-sizing is the choice. Furthermore, both algorithmics and heuristics solutionin the research of dynamic lot sizing are considered, followed by an illustration to provide anefficient algorithm

Self-adaptive randomized constructive heuristics for the multi-item capacitated lot-sizing problem

Capacitated lot-sizing problems (CLSPs) are important and challenging optimization problems in production planning. Amongst the many approaches developed for CLSPs, constructive heuristics are known to be the most intuitive and fastest method for finding good feasible solutions for the CLSPs, and therefore are often used as a subroutine in building more sophisticated exact and metaheuristic approaches. Classical constructive heuristics, such as the period-by-period heuristics and lot elimination heuristics, are first introduced in the 1990s, and thereafter widely used in solving the CLSPs. This paper evaluates the performance of period-by-period and lot elimination heuristics, and improves the heuristics using perturbation techniques and self-adaptive methods. We have also proposed a procedure for automatically adjusting the parameters of the proposed heuristics so that the values of the parameters can be chosen based on features of individual instances. Experimental results show that the proposed self-adaptive randomized period-by-period constructive heuristics are efficient and can find better solutions with less computational time than the tabu search and lot elimination heuristics. When the proposed constructive heuristic is used in a basic tabu search framework, high-quality solutions with 0.88% average optimality gap can be obtained on benchmark instances of 12 periods and 12 items, and optimality gap within 1.2% for the instances with 24 periods and 24 items

Fix-and-Optimize and Variable Neighborhood Search Approaches for Stochastic Multi-Item Capacitated Lot-Sizing Problems

We discuss stochastic multi-item capacitated lot-sizing problems with and without setup carryovers (also known as link lot size), S-MICLSP and S-MICLSP-L. The two models are motivated from a real-world steel enterprise. To overcome the nonlinearity of the models, a piecewise linear approximation method is proposed. We develop a new fix-and-optimize (FO) approach to solve the approximated models. Compared with the existing FO approach(es), our FO is based on the concept of “k-degree-connection” for decomposing the problems. Furthermore, we also propose an integrative approach combining our FO and variable neighborhood search (FO-VNS), which can improve the solution quality of our FO approach by diversifying the search space. Numerical experiments are performed on the instances following the nature of realistic steel products. Our approximation method is shown to be efficient. The results also show that the proposed FO and FO-VNS approaches significantly outperform the recent FO approaches, and the FO-VNS approaches can be more outstanding on the solution quality with moderate computational effort

Modeling Industrial Lot Sizing Problems: A Review

In this paper we give an overview of recent developments in the field of modeling single-level dynamic lot sizing problems. The focus of this paper is on the modeling various industrial extensions and not on the solution approaches. The timeliness of such a review stems from the growing industry need to solve more realistic and comprehensive production planning problems. First, several different basic lot sizing problems are defined. Many extensions of these problems have been proposed and the research basically expands in two opposite directions. The first line of research focuses on modeling the operational aspects in more detail. The discussion is organized around five aspects: the set ups, the characteristics of the production process, the inventory, demand side and rolling horizon. The second direction is towards more tactical and strategic models in which the lot sizing problem is a core substructure, such as integrated production-distribution planning or supplier selection. Recent advances in both directions are discussed. Finally, we give some concluding remarks and point out interesting areas for future research

Mixed integer programming in production planning with backlogging and setup carryover : modeling and algorithms

This paper proposes a mixed integer programming formulation for modeling the capacitated multi-level lot sizing problem with both backlogging and setup carryover. Based on the model formulation, a progressive time-oriented decomposition heuristic framework is then proposed, where improvement and construction heuristics are effectively combined, therefore efficiently avoiding the weaknesses associated with the one-time decisions made by other classical time-oriented decomposition algorithms. Computational results show that the proposed optimization framework provides competitive solutions within a reasonable time

Lotsizing and scheduling problems

Diese Magisterarbeit gibt einen Überblick über Losgrößen- und Reihefolgeplanungsprobleme. Losgrößenplanung und Reihefolgeplanung sind integrale Bestandteile der Produktionsplanung. Sie geben an wie viele Produkte und in welcher Reihenfolge diese produziert werden sollen, um die Gesamtkosten zu minimieren. Der Hauptteil der Arbeit beschäftigt sich mit der Darstellung der unterschiedlichen Problemverfahren. Diese werden anhand von Definitionen, mathematischen Formulierungen und Beispielen dargestellt. Zu Beginn der Arbeit wird das Produktionsplanungssystem (PPS) erläutert (in Kapitel 1.1). Dann werden die Definitionen von Losgrößenplanung und Reihefolgeplanung und ihr Zusammenhang erklärt (in Kapitel 2). Darauf folgen eine allgemeine Problembeschreibung und die verschiedenen Kriterien für Losgrößen- und Reihefolgeplanungsprobleme (in Kapitel 3). Im Hauptteil werden die verschiedenen Problemtypen aufgelistet. Es werden einstufige unkapazitierte (in Kapitel 4) und einstufig kapazitierte Probleme (in Kapitel 5), genauso wie mehrstufige Probleme (in Kapitel 6) und Probleme auf mehreren Maschinen (in Kapitel 7) erklärt. Ebenfalls wird die hierarchische Integration von Losgrößen- und Reihefolgeplanungsproblemen beschrieben (in Kapitel 8). Zuletzt werden ein Vergleich der unterschiedlichen Verfahren dargestellt (in Kapitel 9) und die möglichen Lösungsverfahren (in Kapitel 10) behandelt